$ C = \left[\begin{array}{rrr}1 & 0 & -1 \\ 3 & 4 & 1\end{array}\right]$ $ D = \left[\begin{array}{rr}-2 & -1 \\ 0 & 1 \\ 0 & -2\end{array}\right]$ What is $ C D$ ?
Because $ C$ has dimensions $(2\times3)$ and $ D$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ C D = \left[\begin{array}{rrr}{1} & {0} & {-1} \\ {3} & {4} & {1}\end{array}\right] \left[\begin{array}{rr}{-2} & \color{#DF0030}{-1} \\ {0} & \color{#DF0030}{1} \\ {0} & \color{#DF0030}{-2}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ C$ , with the corresponding elements in column $j$ of the second matrix, $ D$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ C$ with the first element in ${\text{column }1}$ of $ D$ , then multiply the second element in ${\text{row }1}$ of $ C$ with the second element in ${\text{column }1}$ of $ D$ , and so on. Add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{0} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ C$ with the corresponding elements in ${\text{column }1}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{0} & ? \\ {3}\cdot{-2}+{4}\cdot{0}+{1}\cdot{0} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ C$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ D$ and add the products together. $ \left[\begin{array}{rr}{1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{0} & {1}\cdot\color{#DF0030}{-1}+{0}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{-2} \\ {3}\cdot{-2}+{4}\cdot{0}+{1}\cdot{0} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{1}\cdot{-2}+{0}\cdot{0}+{-1}\cdot{0} & {1}\cdot\color{#DF0030}{-1}+{0}\cdot\color{#DF0030}{1}+{-1}\cdot\color{#DF0030}{-2} \\ {3}\cdot{-2}+{4}\cdot{0}+{1}\cdot{0} & {3}\cdot\color{#DF0030}{-1}+{4}\cdot\color{#DF0030}{1}+{1}\cdot\color{#DF0030}{-2}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}-2 & 1 \\ -6 & -1\end{array}\right] $